ANNALS OF PHYSICS 170, 370-405 (1986)
A New Class of Integrable Systems and Its Relation to Solitons
S. N. M. RUIJSENAARS AND H. SCHNEIDER
Mathematics Department, Tiibingen University, Tiibingen, Federal Republic of Germany
Received November 26, 1985
We present and study a class of finite-dimensional integrable systems that may be viewed as relativistic generalizations of the Calogero-Moser systems. For special values of the coupling constants we obtain N-particle systems that are intimately related to the N-soliton solutions of the sine-Gordon and Kortewegde Vries equations, among other ones. 0 1986 Academic Press, Inc.
Contents. 1. Introduction. 2. Discovering the systems. 3. Moser’s argument and its con- sequences. 4. The Lax matrix and the nonrelativistic limit. 5. Explicit solutions: An overview. 6. The relation with soliton solutions. 6A. The first class of solitons. 6B. The second class of solitons. 6C. Soliton space-time trajectories. 6D. A comparison with the IST. Appendix A. Functional equations. Appendix B. Explicit solutions: Proofs.
1. INTRODUCTION
In a previous paper [l] the first-mentioned author developed a scheme to describe directly interacting particles in a relativistically invariant fashion. This scheme was inspired by the discovery of explicit soliton-type S-matrices for some two-dimensional relativistic quantum field theories, the most prominent ones being the massive Thirring model/sine-Gordon theory and the nonlinear O(N) a-models [2-4]. It follows from general principles that if such field theories without particle creation and annihilation really exist, then they should admit a manifestly particle- preserving dynamics, leading to the known S-matrix via wave operators. This con- sideration led to the speculation that similar dynamics might exist at the classical level [ 11. One aspect of this paper is that it validates, combined with further work by the first-named author [5], this speculation in the case of the classical sine-Gordon theory. (The quantization of the relevant particle dynamics, in particular its relation to the field-theoretic S-matrix and bound-state spectrum, will be considered elsewhere [6].) However, the results reported below have a wider scope, as we shall now explain. 370 0003-4916/86 $7.50 Copyright 0 1986 by Academic Press, Inc. All rights of reproduction in any form reserved. A NEW CLASS OF INTEGRABLE SYSTEMS 371
To this end we recall that the Galilei-invariant system of N particles on the line described by the Hamiltonian
w, - Xk) (1.1) is completely integrable when the pair potential equals the Weierstrass function Y(x), important degenerate cases of which are l/sh2 X, l/sin2 x and 1/x2 [7, 8, 91. These systems are commonly called Calogero-Moser systems. In this paper we consider a novel class of integrable N-particle systems that may be viewed as one-parameter generalizations of the former ones. Our systems are Poincare-invariant, in the sense that the time-translation generator H can be sup- plemented with a space-translation generator P and boost generator B such that
{H,P}=O, {H,B}=P, {P,B)=H/c’, (1.2) where c is the speed of light. Specifically, the generators are given by
H=mc2 g ch 0, n f(qj-qk) (1.3) j= 1 k#i
P=mC -f Sh8,n f(qj-qk) (1.4) j= 1 k#i B= -f ,f qi. (1.5) J=l
Here, 0, is the rapidity of particle j and qj the canonically conjugate position, cf. (2.1). Thus, the boost generator B is equal to the freeone for any N, as envisaged in [ 11. However, in Cl] the interaction was introduced in the center-of-mass frame, which leads to a different structure for H and P when N is greater than 2. The latter method leaves much freedom in the choice of potentials, but leads to considerable mathematical (and physical) difficulties. In contrast, the structure (1.3k(1.4) for H and P, with N> 2 and f even, is only compatible with Poincare invariance when f 2(q) equals J + pg(q), integrability being an added bonus. (The relativistic quan- tum dynamics constructed in [ 10, 111, which lead to the S-matrices of the Feder- bush and continuum Ising models [ 121, also have this structure. Admittedly, this is clear only with hindsight [6].) The Calogero-Moser systems may be obtained as special cases of (1.3) in a sense to be detailed in the main text. Here we only mention that they may be thought of as arising in the nonrelativistic limit (that is, when c -+ cc), provided one is willing to let our coupling constants depend on c in a suitable way. For the special case where J. + ,L& reduces to 1 + l/sh2, we obtain a system that is closely related to a class of soliton theories, containing not only the sine-Gordon 372 RUIJSENAARS AND SCHNEIDER
(henceforth sG) theory, but also the Kortewegde Vries (KdV) and modified Kor- teweg-de Vries (mKdV) equations and the Toda lattice. To describe the relation, we take the KdV N-soliton solutions as an example. As is well known, these can be written (up to a constant)
8: ln(det(% + A(t, x))), (l-6) where (t, x) is a space-time point and A an N x N matrix. It can be shown that A has simple spectrum in (0, co). Thus, the eigenvalues of A may be written
&A~.X) < . . . <(p&X). (1.7) The crux is now, that there exists a l-l correspondence between points in our phase space
Qr ((4, 8)EuPNjql < ... qj(t, x)= [exp(tHO-xH’)(q, e)lj,j= l,..., N (1.9) where @ and H’ are two Hamiltonians commuting with the above H and P. Requiring, successively, that q1 (t, x),..., qN(t, x) vanish, we obtain soliton space- time trajectories satisfying x1(t)< ... For asymptotic times these coincide with the N maxima of (1.6). The situation for the other solitons we consider is similar. The sG case has the additional property that p and H’ are equal to H and P, resp., which entails that the trajectories are Poincart-invariant. Also, the equations involved admit a for- mulation as infinite-dimensional integrable Hamiltonian systems, which results from the inverse scattering formalism. As a consequence, the (analogs of the) above A(t, x) can be parametrized with 2N canonical variables (d, r). The sG and mKdV equations are singled out by the l-l correspondence (d, r) + A(& x) + (q, 0) being a canonical transformation. There are previous results in the literature associating particle systems with the above PDE and other ones. These concern the pole motions of rational and elliptic solutions [ 13-15,9] and of soliton solutions or their energy densities [ 16191. These results provided inspiration and encouragement; we shall come back to the relation of our work with that of Bowtell and Stuart [lS] in the main text. Further- more, this paper owes much to previous work on the Calogero-Moser systems, especially by Moser [7] and Olshanetsky and Perelomov [9]. Let us now describe the plan of the paper. In Section 2 we first indicate how we were led to (1.3) with f * = 1 + l/sh* by insisting that the scattering be the same as for sine-Gordon solitons. Supplementing H with the obvious P (1.4) and requiring A NEW CLASS OF INTEGRABLE SYSTEMS 373 (H, P} = 0 then yields functional equations (2.26) that are satisfied whenf’ equals A+ @. For N > 3 these functional equations for the g-function appear to be new. Their validity is proved in Appendix A. The structure of H and P suggests N functions S, ,..., S, as candidates for integrals. In Appendix A we show that these functions indeed commute with H and P; this is a consequence of the functional equations just mentioned. The requirement of involutivity leads to additional functional equations (2.28). We have not proved these for f2 = A + ~9, but their validity for f 2= 1 + a2/sh2 follows from results in Section 3. The crux is that in the latter case every point in our phase space is a scattering state. We prove this by adapting Moser’s arguments leading to the same conclusion for the nonrelativistic l/x2-system [7]. It turns out that this cannot be easily done for the H flow, since Newton’s third law does not hold for it. However, the S1 flow does obey the third law and can be used instead. Since the integrals Si,..., SN reduce to the symmetric functions of exp(8: ),..., exp(6; ) for t + + co (where the 0,* are the asymptotic rapidities), and since the ordering of the particles is preserved, it follows that 8;-,+ 1 = 13,?. This conservation of rapidities then leads to a soliton-type factorization of the phase shifts via a well-known argument. In Section 4 we find an N x N matrix L for the case f 2= 1 + Cr2/sh2, whose sym- metric functions are precisely S, ,..., S,. This hinges on Cauchy’s identity (4.2). Since the spectrum of L is conserved under each of the Sk flows, we refer to L as the Lax matrix. We explain how L is connected with the Lax matrix of the Calogero-Moser systems, and how the integrability of the latter systems may be viewed as a corollary of our results. Section 5 concerns explicit solutions to Hamilton’s equations. It turns out that the positions are related to eigenvalues of certain N x N matrices involving L in precisely the same way as for the Calogero-Moser systems [9]. The technical details are relegated to Appendix B. Section 5 also contains a description of the action-angle map constructed in [S], for the special case c1= 1 arising in Section 6. The latter section concerns the relation to soliton solutions already described above. It is divided into four subsections. We consider two different types of solitons in Subsections 6A and 6B, emphasizing the crucial role Cauchy’s identity plays in leading from Hirota’s formulas to our systems. Subsection 6C concerns the above notion of soliton space-time trajectory and Subsection 6D contains some observations on the relation to the inverse scattering transformation. Appendix A is concerned with the derivation and (partial) proof of the above- mentioned functional equations. In Appendix B we prove some claims made in Sec- tion 5 concerning the solutions to Hamilton’s equations for the S, flow, non- degenerate perturbation theory being our main tool. These results can also be obtained by adapting the methods of Olshanetsky and Perelomov [9], but the details are rather more complicated in our case [ZO]. One more remark: We restrict ourselves in Section 6 to pure soliton solutions, in keeping with the fact that we consider only repulsive interactions in Sections 3-5. As indicated in Section 2, attraction can be introduced via Calogero’s [8] sub- 314 RUIJSENAARSAND SCHNEIDER stitution qj + qj + iy. For N> 2 the resulting systems are considerably more involved, but they can also be handled. For a = 1 the latter systems can be related to mKdV and sG solutions containing solitons, antisolitons and their bound states in a similar way as for the pure soliton case. For more information on this and on relations wih the soliton hierarchies of the Kyoto school [21], see [S]. 2. DISCOVERING THE SYSTEMS A classical relativistic particle on the line is most conveniently described not with its customary momentum p and position x, but rather with its rapidity 8 and the canonically conjugate generalized position q. These are defined by p = mc sh 8, x = q/me ch 8. (2.1) For N free mass m particles the representation of the Poincare group on the phase space R 2N is then generated by Hf= me2 f ch 6, (2.2) j= 1 Pf= mc $ sh %, (2.3 1 j=l Bf= -f ,f qj. (2.4) ,=1 Henceforth, we set m = 1 and c = 1. For N = 2 there exists a simple way to introduce interaction [ 11: Define center- of-mass variables sq1 +q2, cp = it%,+ 021, (2.5) 4’41-42, %rf(%,-e2), and set H=2ch cp(ch %+ V(q, %)) (2.6) P s 2 sh cp(ch 8 + V-(/(4,%)) (2.7) Bs --s (2.8) Then it is clear that the Poincart commutation relations (1.2) hold true, irrespective of the choice of k’. The relative motion is governed by the reduced Hamiltonian H, = ch %+ V(q, %). (2.9) A NEW CLASS OF INTEGRABLE SYSTEMS 375 The case V= V(q) with V a rapidly decaying function was studied in [ 1, Sec- tion 2.23. The scattering is then characterized by a phase shift S(g), where d denotes the asymptotic rapidity. However, it can be seen that for such potentials S(o) can- not equal the phase shift dsG (0) = ln(cth* 0) (2.10) occurring for the soliton-soliton and soliton-antisoliton collision in the sine- Gordon theory. (We have omitted the contraction factor in (2.10), since it dis- appears under the transformation x -+ q, cf. [ 1, l.c.].) A study of the known quantum S-matrix and the paper [18] then led the lirst- named author to consider instead the choice V= ch 13F(q) or, equivalently, H, = ch 8 W(q). (2.11) In this case there does exist a repulsive and an attractive potential leading to the phase shift (2.10). Moreover, these potentials are unique in a sense similar to that of [l, l.c.]. These results were obtained in [22]. Explicitly, the potentials read W,(q) = IcWqP)l (2.12) W,(q) = Ith( (2.13) Before showing how these potentials lead to dso, let us comment on the relation to the work by Bowtell and Stuart [IS]. These authors considered the motion of the poles of the energy density corresponding to the soliton-soliton and soliton- antisoliton solution in the center-of-mass frame. They derive energy functions for their pole particles, whose potentials look superficially similar to ours. However, the parameter y in their energy functions depends on the asymptotics of the motion (it equals l/ch 8). A s such, their description is not Hamiltonian. We shall return to their pole trajectories in Subsection 6C. A quick way to see that the above potentials lead to (2.10) is to note first that (2.11) implies Q* + W*(q) = Hf. (2.14) Now H, is a constant of the motion, so that we may write this for (2.12) and (2.13) resp., Q* + l/sh*(q/2) = E (2.15) 4* - l/ch*(q/2) = E, (2.16) where we have set ErH;-1. (2.17) 376 RUIJSENAARS AND SCHNEIDER Thus, the motion of q is the same as that of a nonrelativistic particle in the Calogero-Moser potential l/sh2(q/2); the attractive case arises through the sub- stitution q + q + in. It is routine to solve the ODE (2.15), (2.16), and to establish that the phase shift equals 21n(l + l/E) in both cases. Now by (2.17) and (2.11) E equals sh2 0, so that the claim follows upon transforming back to the original coor- dinates. We conclude that the functions H= (ch 8, + ch 0,) Wi(ql - q2) (2.18) P= (sh 81+ sh 0,) Wi(qr -q2) (2.19) B= -41 --cl2 (2.20) (where i = r, a) give rise to a Poincart-invariant description of the sine-Gordon soliton-(anti)soliton collision. Note that H commutes with P without restrictions on the potential Wi. Thus one gets integrability for arbitrary potentials (as in the nonrelativistic two-particle case). For N= 3 the situation is different. Here, the clue was provided by a con- sideration of a special three-soliton collision, viz., the symmetric situation where one soliton may be regarded as sitting at the origin. In a particle picture this situation amounts to considering the submanifold q2 = e2 = 0, q, = -q3 = q, 8, = - 0, = 8 of phase space. Thus, one is reduced to a one-dimensional situation, just as in the case of the relative motion of the general two-particle system. Since the phase shift is known from (2.10) and factorization, one can again find the corresponding one-particle Hamiltonian &(q, 0) by a variant of the ‘inverse scat- tering’ ideas discussed above [20]. We shall not spell this out, but rather write down a three-particle Hamiltonian that yields fi(q, 6) upon restriction. It reads H= f shBj n Wr(qj-qk). (2.21) j= 1 k#j The fact that this H yields the desired scattering for the general three-particle system will become clear later. The point to be made here is that the obvious choice for P, viz., to replace ch by sh in (2.21), leads to a function that commutes with H, as desired. Indeed, this follows from a straightforward calculation. Consider now, more generally, the light-cone Hamiltonians (2.22) where f is an even function. Then the requirement that S1 and S, commute is A NEW CLASS OF INTEGRABLE SYSTEMS 377 equivalent to the requirement that f satisfy a functional equation that can be writ- ten f2(4 f(a)f’(a) 1 1 f*(b) f(b)f’(b) 1 =o, (2.23) f"(a+b) -f(a + b)f’(a + 6) 1 as is readily verified. This is satisfied when f’(s) = A+ l@YqL 4 Pea=, (2.24) where 9 is the Weierstrass function (cf. e.g. [23]). Assuming henceforth that (2.24) holds, it follows that the three independent Hamiltonians S,, S-, and exp(8, + 6, + 0,) commute, so that we are dealing with an integrable system. Tak- ing one period of 9 to infinity and choosing 1, and p appropriately, we obtain the previous case, f*(q) = 1 + l/sh2 q. Once one has (2.22), the remaining step 3 + N is not hard to make. This, then, yields the functions H=+(S,+S-,), P=+(s,-s.,) (2.25) of the Introduction, cf. (1.3) and (1.4). From (2.25) it is also clear that the relations (1.2) are satisfied if and only if (S,, S-,} = 0. Calculating the commutator and using evenness off yields as necessary and sufficient condition for its vanishing f aj n f2(4j-qk)=0. (2.26) j=l k#j This functional equation for f is the generalization of (2.23) to arbitrary N. Possibly, it is known that it is satisfied by the &function, but we have not found this in the literature. At any rate, in Appendix A we prove that, more generally, our assumption (2.24) entails (2.26). The physical picture going with the above is clear: Each particle interacts with all other ones, but in contrast to the nonrelativistic case, the kinetic and potential energy terms are bunched together in a product. Taking this picture into account and noting that the function exp(f?, + ... + 19,) commutes with S, and S 1, a natural guess for the higher integrals reads Sk= 1 exp 1 ej nf(qi- qj), k = L..., N (2.27) I c (l,...,N) ( isl > iel 111 =k J41 In Appendix A we prove that these functions indeed commute with S, and S-, , 378 RUIJSENAARS AND SCHNEIDER and hence with H. Moreover, involutivity of the integrals is equivalent to the functional equations 1 (c i$) nf2(qi-q,)=O, k= l,..., N, VN> 1 (2.28) Ic {l,...,N) iGf iEI III=k iCI as is also shown in Appendix A. We have not proved that these equations are valid for k > 1 andf’ = 1+ ~9, but undoubtedly this is true. However, for the degenerate case where one of the periods of the P-function is infinite, the validity of (2.28) is a consequence of results obtained in Section 3. As a matter of fact, we shall restrict ourselves henceforth to the case (2.29) and its limiting case f(4) = Cl + Y2h211’2T YE64 co). (2.30) It should be noted that the corresponding H has the usual cluster property of nonrelativistic Hamiltonians. That is, if the distances between particle clusters go to infinity, then H reduces to a sum of cluster Hamiltonians. This is to be constrasted with relativistic action-at-a-distance schemes, where interaction is introduced in the center-of-mass frame: Here, manifest clustering is generally lost (cf. [ 1 ] and referen- ces), and can only be attained at the price of introducing extremely unwieldy interactions [24,25]. 3. MOSER'S ARGUMENT AND ITS CONSEQUENCES Let us now consider the flow generated by H= f ch6, V,(q) (3.1) j=l on the phase space (3.2) Here we have set I/i(q) E n f(qj-qk) (3.3) kfi A NEW CLASS OF INTEGRABLE SYSTEMS 379 and J is given by (2.29) or (2.30). Hamilton’s equations corresponding to H read cjj = sh 0, Vj (3.4) ei= -Cche,a, v, (3.5) k From this we obtain as the analog of Newton’s second law qj = 1 Ch(8k + ej) vk ak Vj- 4 aj( T), (3.6) k#j where we used vj aj vk + vk ak vj = 0, j#k. (3.7) Thus, the ‘closed system’ condition Cy=, qj= 0 is satisfied precisely when the functional equation (2.26) holds true. Since H is conserved and since ch 8 z 1, there is a finite lower bound on the par- ticle distances and a finite upper bound on the particle rapidities for a given initial point in 9. Thus, the flow is global. Consider now the long-time asymptotics of the flow. Loosely speaking, the forces are repulsive due to our standing assumption (2.29t(2.30). Thus, one expects q,(t)-q,*+tshB,*, ej(t) - e;, t+ fco (3.8) where e,> ... >e- NY e:< ... gj= e”JV, (3.10) 0,= -c eekajVk (3.11) k qj= 1 Fjk (3.12) kfj 380 RUIJSENAARS AND SCHNEIDER where Fik = 2eej+ OkVk ak Vi, j# k. (3.13) Now the desired ‘action = -reaction’ relation Fjk = - Fki (3.14) does hold true due to (3.7). The repulsive character of the forces can be expressed by the inequality F&(q)>O, j>k 91< ... Ljl(-co)> ... >4N(-a)>o, o I-1 ij,-ijk= 1 Fo+ i Fjk+FR. (3.19) j=k j=k+l Here, the first two terms contain the forces due to the particles in the cluster A NEW CLASS OF INTEGRABLE SYSTEMS 381 k, k + l,..., 1; these are positive due to (3.15). The force FR due to the remaining par- ticles may be neglected for t large. (To see this, note that (3.18) and (2.29), (2.30) imply that 1FR 1 = O(e PPEf), IF, 1= 0( t -3), resp., whereas the intercluster forces have a slower decay due to (3.17).) Hence, QI - ijk > 0 for t > T with T large enough, so that 4, - gk is strictly monotone increasing for t > T. Since Q,( co ) - gk (co ) = 0 by assumption, one has Qr - Qk < 0 for t > T. Thus, qr - qk is monotone decreasing, so that the cluster forces are bounded away from 0. But then one has 4, - qk >, 6 > 0 for t large, so that Q,- gk diverges for t -+ co. This contradiction completes the proof. As a consequence, the interparticle distances diverge linearly for 1tl -+ co. Com- bining this fact with Hamilton’s equations (3.10)-(3.11), it is not difficult to see that there exist asymptotic rapidities and positions such that lim (B,(t)-8,+)=0 (3.20) ,+ km lim (q/(t)-q,*--tee?)=0 (3.21) t--t fco Moreover, since the S, are conserved, one must have e,?= 8, ,+ , , j = I,..., N (3.22) and involutivity follows from the fact that {S,, S,} is a constant of the motion (by Jacobi’s identity) with limit 0 for t + &co. We have thus proved that the systems with pair potentials given by (2.29) or (2.30) are integrable in the strong sense defined in [lo, p. 3391. Let us now consider the scattering on configuration space. To this end we paraphrase a heuristic argument that appears to go back to a paper by Kulish [26]. First, the scattering transformation s: (qr )...) qi, 8, ,...) 0,) + (4: )...) q,‘, 0: )...) e;) (3.23) should be canonical as a limit of canonical transformations. Second, writing q&,+, =q,T +A,, (3.24) it follows from (3.22) and the canonicity of S that A, satisfies aye A, = 0, k = l,..., N. Thus Aj depends only on V. But then one is free to determine it by picking 41~ ,..., q; in such a fashion as to ensure that the collisions take place approximately pairwise. Taking the ordering into account, we therefore must have Aj(e-)= - 1 S(e,: -OF)+ 1 s(e,- -e,), (3.25) where s(0) is the two-particle phase shift. In this way, the ‘soliton factorization’ of the total phase shift A, can be understood from (3.22). 382 RUIJSENAARS AND SCHNEIDER Finally, the function s(0) can be calculated as sketched in the previous section. For (2.29) and (2.30) one obtains, resp., 6(0)=p-‘ln(l+a2/sh2q) (3.26) s(e)=o. (3.27) Thus, for the relativistic l/q2-case one gets a ‘billard ball’ scattering, as in the non- relativistic case. The above argument leading to (3.25) is convincing, but not rigorous. In fact, we have not even shown that S is well defined. Elsewhere, the scattering will be recon- sidered in a novel and rigorous fashion [S]. Among other things, it will be shown there that the S-transformation is a canonical bijection from the incoming phase space o- = {(q-, 8p)ER2Nle; > *.. >e;} (3.28) onto the outgoing phase space 52+= {(q+,e+)dPIe: < ... 4. THE LAX MATRIX AND THE NONRELATIVISTIC LIMIT For N = 2 we have Vi = V2 =f(q, - q2). Therefore it is easy to find a 2 x 2 matrix whose trace equals S, = (@I+ e2)f and whose determinant equals S2 = ee’+e2. A sim- ple choice is e.g. the matrix diag(e’lf, e”tf). (A i), where ab is determined by requir- ing 1 - ab = l/f’. Taking this and the B-dependence of the S, as a lead, we make the following Ansatz for the (symmetrized) Lax matrix: Lik = dj Cjk dk ) dj- [ee~Vj(q)]“2, (4-I) where C depends only on q. If we now require that the kth symmetric function of L equal S,, then the 2 x 2 principal minor C(j, k) should equal l/f2(qj- qk), whereas the general principal minor should be a product over all 2 x 2 principal minor con- tained in it. Indeed, if this holds, then the principal minor C(I) knocks out the mutual interactions of the particles il,..., i, E I, and this is what is needed for S, to result, cf. (3.3) and (2.27). A NEW CLASS OF INTEGRABLE SYSTEMS 383 It is not hard to find such a matrix, provided one is familiar with Cauchy’s iden- tity, (4.2) (For a proof, see, e.g., [27, p. 211.) Indeed, this implies that the matrix with elements cjk E (Xj-Yj)1’2(Xj-yk)-‘(Xk-yk)“2 (4.3) has the desired property. Thus, putting xj = (iu + 1) f?q, yj e (iu - 1) ep4/, (4.4) which implies -1 Ch~(~j-gk)+iaSh~(q,-gk) 1 , (4.5) it follows that I-‘. (4.6) As a consequence, we can ensure that the functions Si,..., SN, with f given by (2.29) are equal to the symmetric functions of L, provided we require (1 +a2)-‘=a2. (4.7) Setting CI = ,uy/2 and taking ,D to 0, we obtain a Lax matrix for the case (2.30). We are using the term ‘Lax matrix’, since it is obvious from the above that the spectrum of L is invariant under the flow generated by H (and also under the Sk flows, for that matter). We shall now clarify the relation of our L with the Lax matrix L,, of the Calogero-Moser systems with pair potential l/sh’ x and l/x*, and show how the integrability of the latter systems follows from our results. We begin by noting that we are free to replace 8, in L by Bej with BE (0, co), since this amounts to a scaling. Then we get, collecting the above results, ’ ,vjf(qj--41) n f(qkbqm)]la (4.8) m#k 384 RUIJSENAARS AND SCHNEIDER where cjk =r/[r + i(qj-qk)l (4.9) when f= [ 1 + y2/q2]“2, Y E (0, CQ1, (4.10) and where ch~(q,-qk)+iash~(qj-qq,) 1 -1 , .2,t,-2- 1 (4.11) when f=[ 1 + a2/sh2 (7 )I112 , a,PE (0, CfJ1. (4.12) Henceforth we shall refer to these two cases as case I and case II. Setting now Y =gA a-g/3 (4.13) it is clear by inspection that L is holomorphic at p = 0 and satisfies L=Q +pL,,+o(p2), p-0. (4.14) Here, L,, is the nonrelativistic Lax matrix [7-91 Lpw(q, e)jkEejjSjk--g(1--jk)h(qj-4k) (4.15) with l/q (case I) h(q) = (4.16) (case II) As a corollary, it follows that the symmetric functions of L,, commute. To see this, consider Ps(il)~(p-L(L(8)-n)-;111) = f (-A)‘F,(/?) (4.17) /=O where s&k@). . (4.18) A NEW CLASS OF INTEGRABLE SYSTEMS 385 Now PB(A) is holomorphic at B = 0; in fact, by (4.14), P,(A)=IL,,-19(= 5 (-A)‘S,-I,“,. (4.19) I=0 Hence, one has SLnr= jimo FN--I(8), (4.20) so that the S,,,, are uniform limits of linear combinations of the commuting Hamiltonians S,(a). Therefore the S,,nr commute, as claimed. The above arguments are simple and straightforward from a mathematical point of view, but they may leave a physicist unsatisfied: If one picks B # 1, one can no longer view 8 as a particle rapidity in the sense of (2.1). However, it is also possible to get the above results from a nonrelativistic limit, provided one is willing to pay a mathematical price of some additional complication and a physical price of having the coupling constants depend on the speed of light. Indeed, let us take Y -+ g/c, u --, glc, P + PIG (4.21) and transform to variables xi, pj by using (2.1) with m = 1. Then it is readily verified that UdXYP), @(PI)= 1 + c-l L,(x,p) + o(c-2), C-+03 (4.22) Moreover, replacing H by H-NC’ and taking c to cc, the Poincare generators (1.3)(1.5) converge to the Galilei generators H,,,= f $+g2 1 h2(xj-xk) (4.23) j=l l (4.24) j= I B,, = - ‘f x, (4.25) j=l We would like to add that we do not see a physical reason why the coupling con- stants should depend on c. We recall in this connection that for the special value CI= 1 the scattering of our particles is the same as that of sine-Gordon solitons. The latter scattering is independent of the value of c in the sG equation. In fact, the sG theory appears not to have a sensible nonrelativistic limit, just as our systems do not have a nonrelativistic limit when one keeps the coupling constants fixed. 386 RUIJSENAARS AND SCHNEIDER 5. EXPLICIT SOLUTIONS: AN OVERVIEW Let us denote by x, (I),..., xN(f) the solutions with initial value (x,p) to Hamilton’s equations for H,, , given by (4.23). As shown by Olshanetsky and Perelomov [28,9] for the case h(x) = l/x, these solutions are equal to the eigen- values of the matrix X+ t L,,(x,p), where X- diag(x,,..., xN) and where L,, is the nonrelativistic Lax matrix (4.15) with h(x) = l/x. This may be viewed as a limiting case of another result obtained by these authors [29,9] for the case where h(x) in (4.23) and (4.15) equals l/sh(px/2): Then the quantities exp(pxJt)) are equal to the eigenvalues of the matrix exp(,uX/2) exp( t$,,) exp(&V/2). With the proviso that one does not take the H flow but the S1 flow, this state of affairs persists for our more general cases I and II, resp. We shall present precise statements and detailed proofs in Appendix B. Here, we want to isolate one of the main ingredients of the proofs, consisting in the use of nondegenerate perturbation theory. To this end, consider first a matrix of the form E, = Q + tL, (5.1) where Q-diag(q,,..., qN), q1 < ... gj(o) = L,y (5.3) by first- and second-order perturbation theory, resp. Next, consider a matrix of the form E,+;Ql2 e'L eQ/2 (5.5) For ItI small enough we may denote the eigenvalues of E, by eY’(‘),..., eyN(‘). Moreover, by nondegenerate perturbation theory one has &(‘) = ee + te%‘2LJJ .&II2 1 eqJJ2Ljk eqkLkj @d2 + t2 - eqJ12(L2),ii&J2 + C 2 e4 - p k#j 1 + O( t3) =e’ l+t&+g L,?‘+ c L&kjCth~(C&-qk) +o(t3) , k#j > 1 t +0. (5.6) A NEW CLASS OF INTEGRABLE SYSTEMS 387 Hence, it follows that The point is now, that our Lax matrix L(q, 13) for case I with /? = 1 and for case II with /? = p = 1 (cf. (4.8t(4.12)) is such that (5.3)(5.4) and (5.7)(5.8), resp., can be written 4;(O) = eqVj (5.9) iii(O)=2 1 eeJ+ek vk dk vj, (5.10) kZ/ as a straightforward calculation shows. Comparing this with (3.1OF(3.13), we see that we have verified a necessary condition for q1 (t),..., qN(t) to be the solutions to Hamilton’s equations for S, with initial value (q, (3) E Q. Complete proofs require more work and are relegated to Appendix B. To prepare the ground for the next section, we now summarize some further results from [S], pertaining to case II with tl = 1. Fix (q, 0) E Q and consider L defined by (4.8) (4.1 l), (4.12), with /? =/J = 1 and a = 0. Then there exists a unique orthogonal matrix 0 such that OLO ~ 1 = diag(e”,..., eBN) (5.11) (OeQOpl),, = e4fi2( Vj(0))1’2 ’ eikj2( V, (Q))l” (5.12) ch l/2(8,- 4,) Here, (8, e) belongs to the action-angle phase space sis {(ij, c?)ER2Nldl < ... -d,}, (5.13) and the map (q, 0) -+ (4, d) is a canonical bijection from Q onto fi. This map linearizes any Hamiltonian of the form H,= Trf(ln L), fog@) (5.14) in the sense that the nonlinear flow (4, 0) --) (q(f), e(f)) = ew(tfff)(q, 0) (5.15) on Q corresponds to the linear flow (44 + (61+ ti’uJ1L.., !?N+ f&r), 0) (5.16) on B [SJ. 388 RUIJSENAARS AND SCHNEIDER Setting (5.17) it now follows from (5.12) that eQtt) is similar to the matrix A(t) with elements (5.18) Using (5.11) one also concludes that eQ(‘) is similar to eel2 etf’(lnL) eQi2; note that this amounts to the previous result for the S, flow when f(u) = e”. Moreover, since S-i = S,_ , S; l maps into the Hamiltonian c,“= 1 e-4 on fi, it corresponds to the function e - ‘, so that H and P correspond to ch u and sh u, resp. Thus, one has S+l=TrL”, H=Tr((L+L-‘)/2), P=Tr((L-L-‘)/2). More generally, one can consider multi-parameter flows (5.19) on B, which map into flows (5.20) on 8. Modifying (5.17)-( 5.18) in the obvious way, one obtains matrices Q(t, ,..., t,) and A( t, ,..., t,) such that eQ and A are similar. The soliton solutions considered in the next section can be written in terms of the positions qj(t, x) corresponding to a function tp - xH’, Hj= H/,, j=O, 1, (5.21) where f0 and f, depend on the case at hand; for lattices one should take x E E. We shall now turn to the details. 6. THE RELATION WITH SOLITON SOLUTIONS The soliton solutions we consider here have all been obtained by Hirota some fif- teen years ago. (His solutions and other ones are reviewed in [30].) The solitons that we shall relate to our l/sh2-system with o! = 1 fall into two classes, discussed in Subsections 6A and 6B. In Subsection 6C we introduce and discuss a notion of soliton space-time trajectory, while Subsection 6D concerns a comparison with the inverse scattering transformation. A NEW CLASS OF INTEGRABLE SYSTEMS 389 6A. The First Class of Solitons The N-soliton solutions of this type can be expressed in terms of a function In r, where T is of the form c exp f BjkPjPk+fPjtj . (6.1) /I ,,..., &v=O.l ( j-ck i > Here one has et?, &vi(#), (6.4) we can rewrite (6.1) as det(Q + A) (6.5) where Ajk-exp[$ijj+ tp(dj)-xa(6j))](vj(t?))“’ ’ ch +(oj-&) (6.6) ‘exp[$(ijk + tp(dk)-xa(d,))]( vk(6))“‘. To see this, note that 121+ Al equals the sum over all principal minors of A and observe that the principal minor A(i,,..., ir) equals the term in (6.1) with pi ,,..., pi, equal to 1 and the remaining ,B’S equal to 0. Indeed, we have already seen in Sec- tion 4 that = fl th2 ;(e,- 8,) (6.7) i-ck cf. (4.5)-(4.6). After these substitutions we are therefore in the situation sketched at the end of the previous section: The functions so(u) and fi (u) are equal to primitives of p(u) and a(u), resp. Moreover, since A is similar to eel we have 111+ A 1 = IQ + eQJ, so that we may write In r = $J ln( 1 + e*(r3x)) (6.8) j=l 390 RUIJSENAARS AND SCHNEIDER We shall consider two special cases. The first one is the KdV equation. Hirota [31] writes it as ti - 62424’+ IA”’ = 0 WV) (6.9) and shows that U= -2(ln r)” solves (6.9) when r is given by (6.1)-(6.2) with p = a3 (6.10) Oj-a, ’ eXp(Bjk)= - . ( aj+a, 1 Furthermore, one should take O a=,‘, p = e3’ (6.13) to obtain the desired form (6.3) for the phase factor. In this case we can take fi (u) = e”, f0 (u) = e3”/3, so that H’=TrL, P=$Tr L3. (6.14) The second special case is the Toda lattice. In this case there is also dependence on whether one is dealing with a right-moving (E = 1) or a left-moving (E = - 1) soliton. Specifically, one has [32] p = 2~ sh (a/2) (6.15) (6.16) Moreover, one should take x E Z in (6.2). Here, the substitution that yields (6.3) reads e-‘, E= 1, e>o th;= (6.17) e’, &=-1, e a = 2 arsh PC2 (6.18) sh 8’ from which the functions f, and Jo can be determined. Before turning to the second class, we should point out that the fact that for the above two cases 7 can be written as a determinant is well known. In fact, it is in this A NEW CLASS OF INTEGRABLE SYSTEMS 391 form that r naturally appears when the pure soliton solutions of the KdV equation and Toda lattice are obtained via inverse scattering (see, e.g., [33]). We have star- ted from (6.1), since there are cases where soliton solutions of this form are known, but where it is not at all clear whether they can be written in terms of a determinant. (A case in point are the Boussinesq solitons [34].) The above clearly shows that the coincidence of the two expressions (6.1) and (6.5) hinges on Cauchy’s identity (4.2). Similar remarks apply to the solitons that will be considered next. 6B. The Second Class of Solitons The second ty$e of N-soliton solutions can be expressed in terms of a function arctg( g/f ), where g= co ew Bj/cPjPk + C PjCj (6.19) P, >....PN = 0, 1 i > (6.20) Here, Co and 1’ denote summation under the condition that C,“= 1 p, be odd and even, resp., whereas the other quantities have the same features as in (6.1). Suppose that the substitutions p = p(e), 0 = o(d) yield exp(B,,) = - th2 +(ei - 4,). (6.21) Then we claim that if we also substitute (6.4) into g and A we can write arctg( g/f) = Tr(Arctg A), (6.22) with A given by (6.6). To prove this, we first point out that (6.19)-(6.20) can be rewritten as 1 g=i(lQ+iA[-[Q-iAI) (6.23) f=&(jQ +iAl + 111-iAI). (6.24) Indeed, this follows in a similar way as the equality of (6.1) and (6.5). Invoking now the well-known formula IQ + TI = exp -n~l~Tr(~)], lITI < 1 (6.25) and the power series of the Arctg-function, (6.22) readily follows. Combining (6.22) with the similarity of A and eQ, we obtain the desired analog of (6.8 ), viz., arctg( g/f) = ; Arctg(eY~“~“‘). (6.26) j= 1 392 RUIJSENAARS AND SCHNEIDER We consider three special cases. For the modified KdV equation d + 240~~ + v”’ = 0 (mKdV) (6.27) Hirota [35] shows that v = (arctg(g/.I)) ’ is a solution, provided one sets p = O3 (6.28) (6.29) Thus, the KdV substitution (6.13) yields the desired result (6.21), and therefore (6.14) holds true for the mKdV case, too. For the sine-Gordon equation cp” - @= sin cp (sG) (6.30) Hirota [36] obtains N-soliton solutions of the form cp = 4 arctg(g/f) by taking a=(1 +py (6.31) oj-ak+pj-pk * exp(Bjk) = - (6.32) ( aj+uk+pj+pk > Thus, the substitution p=sh& a=chd (6.33) yields (6.21). Consequently, one gets in this case (6.34) The third case consists of a system of network equations, which Hirota [37] reduces to an equation for a function cp(t, n) with n E Z. He finds N-soliton solutions of the form cp = arctg(g/f), where the dispersion relation between p and (T and the phase factor are given by (6.15) and (6.16), resp., except that eB has an additional minus sign. Thus, the substitution (6.17) leads to the desired result, and (6.18) holds for this case, too. As a consequence, these solitons are related to the Toda solitons in the same way as the mKdV solitons are related to the KdV solitons. 6C. Soliton Space-Time Trajectories From the relations (6.8) and (6.26) it is evident that the above N-soliton solutions may be viewed as linear superpositions of N ‘single-soliton’ terms. Moreover, these formulas suggest a natural notion of soliton space-time trajectory A NEW CLASS OF INTEGRABLE SYSTEMS 393 for the KdV, mKdV and sG solitons. The point is, that if we fix t, E R, then there exist uniquely determined x1 (to),..., x,(t,) E R such that qj(klY Xjhd) = 0, j= l,..., N. (6.35) (Note that the value 0 is the obvious one for a single soliton.) To show this, we recall that the space-translation generator H’ equals S, in the KdV and mKdV cases, and equals P in the sG case. Thus we have qj = -e’lV,(q) ((m)KW (6.36) q; = -ch 0, V,(q). (sG) (6.37) Since these derivatives are negative and bounded away from 0, existence and uni- queness of the trajectories (6.35) follows. Moreover, one has X,(f)< ..’ Thus the eigenvalues of A are given by exp(q;(t,x))=exp(-xch@[af(~~-l)“~] (6.41) where we have put a E cth t? ch( t sh 6) (6.42) Since our soliton trajectories are defined by (6.35), they read in this special case x,(t)= T--$ Archa, (6.43) 2 with a given by (6.42). These trajectories coincide with the pole trajectories found in [18]. (Mutatis mutandis, this also holds for the center-of-mass soliton-antisoliton collision.) However, already for N= 2 there appears to be no sensible notion of ‘pole trajec- tory’ as soon as one leaves the center-of-mass frame [20]. 6D. A Comparison with the IST Let us conclude with some remarks concerning the relation to the soliton action- angle variables arising via the inverse scattering transform (ET). A particularly useful source for our purposes is the contribution of Flaschka and Newell in [39]. (For the KdV, sG and Toda cases, cf. also [40], [41] and [42], resp.) When one restricts the IST to N-soliton solutions, it may be viewed as a canonical transformation to a system of N particles on the line, with action-angle variables r, ,..., rN, d, ,..., d, that are suitable functions of the scattering data of the linear eigenvalue problem. However, this Hamiltonian system is free: The knowledge about the interaction that gives rise to the phase shifts is coded in the IST. What we have shown is that there exist interacting Hamiltonian particle systems that lead not only to the same scattering, but that also have an immediate relation to the solutions involved, as expressed by the formulas (6.8) and (6.26). The canonical map from 52 onto b described at the end of Section 5 is the analog of the IST for our particle system, the variables g,,..., I!?,,,, qi,..., qN being the action- angle variables. For the KdV and Toda cases the map (d, r) -+ (q,8) is not canonical, as may be seen from the references cited above. However, for the mKdV and sG cases the correspondence is not only canonical: It is in essence (i.e., up to trivial scaling and ordering changes) the identity. This can be seen from [39, p. 377 and p. 4053: The crux is that up to numerical factors the function (6.44) equals the residue of a(i))’ at the pole i = ii. ANEWCLASSOFINTEGRABLESYSTEMS 395 APPENDIX A. FUNCTIONAL EQUATIONS In this appendix we first prove the equivalence of the functional equations (2.28) to the involutivity of the functions S,,..., S, (Theorem A 1). This results in the equivalence of the special case (2.26) to {H, P} = 0 and to S, ,..., S, being integrals for H (Corollary A 2). Finally, we prove (2.26) for f2 = A + ,uY (Theorem A 4), using Lemma A 3. We begin by introducing some notation. Let Z, J be disjoint subsets of (l,..., N} and let f be even. We set W)= iGln f(ql - q,)= (.m (Al1 jsJ (I) = (II”) = (r’) (A21 o,= 1 Q,, a,= c aq, (A31 icl rel and note the relations (I4 = (1, mz2.Q I, v z, = z, I, nZ,=@ (A4) (a,+ aJ)(z.z) = 0. (A51 With this notation the functions (2.27) can be written Sk = C e”(Z), k = I,..., N. (A6) 111 =k We also define Sk= c e-“(z), k = - l,..., -N. 647) THEOREM Al. One has (Sk, s,> =O, t’(k, 1)~ { l,..., N}‘, VN> 1 (A8) if and only if c d,(Z)‘=O, VkE {l,..., Nj, VN> 1. (A9) (/I =k Proof. We begin by noting that S, commutes with S, ,..., S,.- , . Moreover, since (I) = (I”), we have Sk--==,&‘, k = l,..., N- 1 (A101 396 RUIJSENAARS AND SCHNEIDER Thus, (A8) is equivalent to {Sk,S-l} =o, V(k, I) E { l,...) N- l}‘, VN> 1. (All) Now we have {Sk, L} = 1 {W), e@J(J)) 111 =k IJI =I =- c e”‘-e’[(r) a,(J) + tJ) aJ(l)l (A121 111 =k IJI =I Hence, setting ArT\J, BrJ\Z (A131 CrInJ, DE (Iu J)’ and using (A4) and (A5) we obtain {sk?s-,}=- 1 e ‘,+AB)*(A LJ B) &(CD)*. (A141 111 =k IJI = I This vanishes if and only if the coefficient of eBA~ BEvanishes for any fixed A and B. Hence, the equivalence of (Al 1) and (A9) follows. 1 COROLLARY A2. The following three relations are equivalent: {fv)=O, VN>l (A151 (H, Sk} =o, Vk E (l,..., N}, VN> 1 (‘416) ,$, d,({i})‘=O, VN> 1 (A17) Pr%f. We recall that H=$(Sr+S-,), PE~(S,-S-,) and note that {S,, Sell = - fj a,(fi>)’ (‘418) i= 1 Thus we need only show (A17) implies {Sl, L> =o, VZE {l,..., N- l}, VN> 1 (A191 But this follows from (A14), since ICI < 1 for this commutator. 1 A NEW CLASS OF INTEGRABLE SYSTEMS 397 In the next lemma it is convenient to use the notation (ij) -f(q, - qj)2. Then the 1.h.s. of (A17) reads f ai n (ii). (A201 i=l Jfi LEMMA A3. Assume that (A20) vanishes, VN > 1. Then (A20) also vanishes if (ij) is replaced by (ij) + Lj, ,lj E 62. Proof. Performing the indicated substitution and expanding the products, the result can be written IJIGN--I icJT kfi kcJC where )lJ-njcJ)lr. Since the coefficient of lJ is of the form (A20), the claim follows. 1 THEOREM A4 Assume that (A221 where .!Y is the Weierstrass function. Then one has ? aj nf2(qJ-qk)=oy VN> 1. (A23) j= I kfj Proof Using the notation of Whittaker and Watson [23], we pick w, equal to rc and q E eiw E e --‘, Rex>O. (~24) (This can be attained by scaling.) Then we have from [23, p. 460, Ext. 351: 1 F(u)r-- .f n 2n q2” (einu + e-inu) 4sin’tu n=, l-q 6425) =9(u) + c, UER (A261 with a certain constant c. Thus, by Lemma A3 we need only prove (A23) for f * = F. To this end, we begin by noting that for c1> 0 1 -1 in(u+ ia) 4sin2~(u+ia)=e-‘“+“(1 -eiu-~)2=-n=l fne . (A27) 398 RUIJSENAARS AND SCHNEIDER For z in the strip 0 < Im z < 2 Re x we may therefore write -F(z)= f n ein=+& [einz+e-inz] II=1 ( > =.iy, .&ezn2 =2 5 nr”“eim, (A281 n= --co sh nx Here we used (A24) in the last step, and the prime signifies that the term with n = 0 is to be omitted. Next. we set zjs u, + iaj, j = l,..., N (A29) where o F(z,-:,)= -2 ft ne”‘.i-k’nrein(r,~;k) n=--oo shnx where E( . ) denotes the sign function. Consider now the function G(ul ,..., uN)s ‘f a,, n F(u,-u,+i(aj-a,)). (~32) j= 1 kfj By analyticity, it suffices to prove that G = 0. Since G is a smooth function on the torus TN = [0, 27rlN, we need only show that all of its Fourier coefficients G(n) = lTNdu ein’UG(u), nfzZN (A33) vanish. To this end we use integration by parts and (A31) to obtain 6434) exp[c(j-k) mkx+ im,(uj-uk)-mk(aj-ak)]. Here and below, - denotes proportionality. We observe that the integrations and summations may be interchanged, since the series involved converge uniformly. A NEWCLASSOF INTEGRABLESYSTEMS 399 To prove that e(n) vanishes, we first dispose of the special case where n/=0 for some ZE {I,..., N}. Then the term Tj with j = 1 clearly vanishes. Now consider T, with j# 1. The r+dependence of Tj is given by the factor exp( -im,u,) with m,# 0. Hence, doing the urintegral first, one sees that T, = 0. Thus, it remains to show that e(n)=0 when nE h*N. To prove t hi s, we do the integrations and pull out con- stants, obtaining G(n)- 2 nj6,, n -exp[&(j-k)nkx-nk(cri-crk)]nk j=l k +j sh nkx (A35) - &I,~$~ sh njx fl exP[s(j-k) nkxl k#j where (T = C;“= I n,. Writing this out, we get (n,+ “‘+n,“-,).Y nN.x-,-nN.Y f3n) - 60, Ce (e 1 + e(“, + “. +n&2)r nN-,r- e-“‘+-,” (e ) e --nN-y + . . . + (e”l”- e-“l”) e’-“2”’ -w-y = Soo(eur - e-““) = 0. (A36) Thus, the theorem is proved. 1 Combined with Corollary A2, this yields involutivity of S1 ,..., S, for N < 4. Since we have not proved (A9) for k > 1 andf* = A + ,up’, complete integrability for N> 4 is open. The validity of (A23) for f” = 1+ p/sh’ follows from Theorem A4 by taking a suitable limit. This case can also be handled directly by proceeding as in the above proof, using the fact that the Fourier transform of l/ch2(xq/2) is proportional to 8/sh 0. Of course, we know more in this case: It follows from Section 3 that s I,.*., S, are in involution, so that the functional equations (A9) hold when f==A+ @h=. Using the substitution q + iq, the same result follows for f 2= A+ p/sin2. APPENDIX B. EXPLICIT SOLUTIONS: PROOFS In this appendix we state and prove two theorems concerning the solutions to Hamilton’s equations 4, = Be"'vj (q), j = l,..., N (Bl) e,=- f eaeka.j vk bid, j = l,..., N U32) k=l 595/170/2-10 400 RUIJSENAARS AND SCHNEIDER for the flow generated by the function Sl(q, 19)s f eP%Vj(q) 033) j=l on the phase space sz= ((4, 8)EFPNlql< ... These results were described in an informal way in Section 5. In this appendix it is convenient to write our Lax matrix (4.8) as Lj~ = dj Cj~ dk 035) dj(q, 0) - eBej12(Vj(q))l12. @6) We recall that C is given by (4.9) and (4.11) for the cases I and II, resp. THEOREM Bl. (Case I) Let (q, 6) E R and set (B7) where Q(q) = diag(ql,-., qN). 038) Then the spectrum of E, is simple and real for all t E R, and the eigenvalues 41(t)< ... < qN(t) satisfy djCt) > O, j= l,..., N, vtgllx. (B9) Setting ej(t) E 8-l lnC4ji(t)lBVj(dt))19 j= l,..., N, (BlO) the functions qj(t), Oj(t), t E R, solve Hamilton’s equations (Bl) - (B2) with initial value (4, e). THEOREM B2. (Case II) Let (q, 0) E Q and set 4 = exr@QP) expMW ewW/2). Wl) Then the spectrum of E, is simple and positive for all t E R and the eigenvalues e/4~1u)< . . . < ew7Nu) satisfy (B9). Defining Oj(t) by (BlO), Hamilton’s equations (Bl)-(B2) with initial value (q, 0) are satisfied by qj(t), f?,(t), VtE R. A NEW CLASS OF INTEGRABLE SYSTEMS 401 Proof of Theorem Bl. We shall first prove the claims for t E ( - 6, a), where 6 satisfies 0 < 6 < co. By exploiting that S1 commutes with H, we shall then show that one must have 6 = oz. First, let 8 be the largest number such that a(E,) is simple, Vt E (- & $). Clearly, for such t the eigenvalues of E, are real and smooth, and one can find a unitary U,, which is continuous in t, satisfies U, = II and is such that US, U;’ = diag(q, (t),..., qdf)) = Qt, tE(-i7, S). (B12) Moreover, these requirements determine U, uniquely up to a unitary of the form diag(u, (t),..., ~~(1)) with uj continuous and u,(O)= 1. Second, set d(t) E U,d, tE(-&9) (B13) and let 6 < 8 be the largest number such that d,(t) f0, j= I,..., N, VlE(-S,S). (B14) (By continuity, 6 > 0.) Then we can render U, unique for 1tl < 6 by requiring d,(t) >O, j = l,..., N, lE(-S,S). (B15) (This fixes the phases u,,..., uN mentioned above.) For later use we note that U,lTd=d(t), u316) since U, is unitary and d(r) is real. Third, we observe that (4.9) implies i[E,, L] = yd@d- yL. (B17) Hence, setting L,=U,LU;’ (Bl8) and using (B12), (B13) and (B16), we obtain XQl, &I = yd(t)Od(t) - YL,. (B19) Solving this for L, yields (Lt)i/c=Ydj(f)CY +i(qj(t)-qk(t))l-’ dk(t). WO) We are now prepared to prove that gj > 0 and that (Bl )-(B2) hold for ItI < 6. To this end, fix t, E ( -6,6). For h small enough we may write ut, Et, + h ut, ’ = Q, + hB&, 3 0321) 402 RUIJSENAARS AND SCHNEIDER where we used (B7), (B12) and (B18). Hence the eigenvalues of the r.h.s. are equal to q1 (to + h),..., qN(tO + h). From perturbation theory it then follows that at time t, we have ljj=/?Lii=# W2) iii = 2k;j (BLjk)(BLkj)(qj-qk)-l 2 (~23) = 2/P c dfJ d2k k#j cY’+ (qj-~k)21(qj-qk)’ Here we used (B20) in the second step. Moreover, since to is arbitrary, (B22k(B23) actually hold for all t in ( - 6,6). Combining (B15) and (B22) one sees that Qj > 0, so that one has, using (BlO), dj(t)* = epe~“‘Vj(q(t)), V’tE(-&6). (~24) Hence, (B22)-(B23) can be written, using also (4.10) tjj=2fi2 C eP(ek+B,)l/kakv, I’ 0326) k#i Differentiating (B25) and using (B26) and (3.7) now yields (B2). Thus, Hamilton’s equations for the S1 flow are satisfied for 1tl < 6. Finally, assume that 6 < co. Since Si commutes with H, there is a finite upper bound on the rapidities IO,(t)1 and a finite lower bound on the distances qj+ 1 (t) - qj( t) for t E ( - 6, 6). In view of (B24) this implies dj( t) z E > 0, j= l,..., N, VtE(-&is). (~27) Hence, it follows that 6 equals 3. But then o(E~) or o(E-,) is not simple, which contradicts the lower bound on the particle distances. 1 Proof of Theorem B2. For notational convenience we only consider the case /I = p = 1, the general case being obvious from this. We first take c(E (0, 11. Then a E [0, co), so that L is self-adjoint, cf. (4.11). We can therefore proceed as in the previous proof. As the analog of (B12) we get U,E, U;’ = diag(eylcr),..., eqN”‘) (J-Q8 1 -e Qt 3 tE(-S,S). We then define d(t) by (B13) and render U, unique by requiring (B15). Then (B16) follows again. A NEW CLASS OF INTEGRABLE SYSTEMS 403 The third step is more laborious here. Let us introduce y, E e - 812~~ - lLe -Q/2 9 (~29) the point being that by virtue of (4.11) we have ia[E,, Y f ] = ia(eQ/2Le-Q/2 - e-Qi2LeQ12) = 2d@ d- (eQi2~e-Q12 + e-Q12~eQ12 1 =2d@d-(E,Y,+ Y,E,). (B30) Thus, if we set ,r, G eQ~12u, y u- 1 ,Q[l2 f f 9 (B31) then we may conclude that ia(eQd2L, ~~QID-~-QII~L, eQd7-) = iaU,[E,, Y,] U,-’ =2d(t)@d(t)-(eQd2L,e-Qi’2+epQd2L,eQr’2), (~32) where we also used (B28) and (B16). Hence, we get the desired analog of (B20), (L,)jk=dj(t)Ccht(qj(t)-qg,(t))+iash ~(qj(t)-qk(t))lp’ dk(t). (B33) To exploit this as in the previous proof, we first combine (B31) and (B29) with (B28) and (Bll) to obtain L; = eQr12~t e-Q12~2 e-tL e-Q12~; I eQr12. By induction and continuity we then infer that ehL,~eQt/2~~e-Q12e~h-~~Le~Q12~~1eQ,/2 f (B35) Thus we may write for t, E (-6,6) and h small enough, eQro/2 ehLlo eQ1012 = ,TJ eQ/2e(ro+h)LeQ/2 u-1 10 hl (B36) where we used (B28) and (Bll). It follows from this that the matrix E,,, h is similar to the matrix eQto’2 ehLroeQ@j2. Thu s, we are in the position to invoke perturbation theory as in Section 5 (see (5.5))(5.10)). From this our assertions then follow in the same way as in the previous proof, so that we omit the details. It remains to deal with the case IXE (1, cc ), taking e.g. ia > 0. Then L is not self- adjoint, but now L is real, so that E, is real. Thus, a(E,) is symmetric w.r.t. the real axis. Fixing ME (1, cc) and noting that a(E,) = a(eQ) is simple and positive, it follows that o(E,) is simple and positive for 1tJ small enough. This observation enables us to proceed as in the preceding case. Here, we can require that U, be 404 RUIJSENAARS AND SCHNEIDER invertible and real for ItI < 8. Also, defining d(t) by (B13), we let 6 < 8 be the largest number such that d,(t) > 0, (U,‘Td)j>O, j= l,..;, N, VtE(-d, 6). (B37) (Note that 6 > 0 by continuity and reality.) We can then render U, unique by requiring in addition U,‘Td= d(t), WE(-&h). (B38) To prove this, we note that the previous requirements determine U, up to a matrix diag(c, (t),..., cN(t)), with cj positive, continuous, and such that ~~(0) = 1; in view of (B37) and the relation d(t) = U,d the extra requirement (B38) fixes these functions. After these preliminaries, the proof for the case tl E (0, l] applies verbatim. 1 ACKNOWLEDGMENTS S. R. has benefited from conversations with M. Adler and P. van Moerbeke. Note added: Meanwhile, some external field couplings preserving integrability have been found [43]. Moreover, relativistic generalizations of the Toda systems have been discovered and partly solved [44]. Also, a Lax matrix has been found and involutivity has been established for the general B-function case; the functional equations (2.28) expressing classical integrability turn out to follow from functional equations that imply quantum integrability [45]. REFERENCES 1. S. N. M. 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